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Second-order logic and foundations of mathematics. (English) Zbl 1002.03013

Summary: We discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
03B15 Higher-order logic; type theory (MSC2010)
03E99 Set theory
03A05 Philosophical and critical aspects of logic and foundations
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References:

[1] Acta Philos. Fenn. 8 pp 57– (1955)
[2] Proceedings of the international colloquium in the philosophy of science, London, 1965 1 pp 138– (1967)
[3] DOI: 10.1016/0003-4843(72)90002-2 · Zbl 0248.02061 · doi:10.1016/0003-4843(72)90002-2
[4] Model-theoretic logics pp 599– (1985)
[5] A decision method for elementary algebra and geometry (1948)
[6] Subsystems of second order arithmetic (1999) · Zbl 0909.03048
[7] DOI: 10.1007/BF02761238 · Zbl 0532.03014 · doi:10.1007/BF02761238
[8] DOI: 10.1007/BF02761237 · Zbl 0532.03013 · doi:10.1007/BF02761237
[9] Foundations without foundationalism (1991) · Zbl 0732.03002
[10] The principles of mathematics revisited (1996) · Zbl 0869.03003
[11] Formal systems and recursive functions (Proceedings of the eighth logic colloquium, Oxford, 1963) pp 131– (1965)
[12] Essays on the foundations of mathematics pp 269– (1961)
[13] Theory of models (Proceedings of the 1963 international symposium Berkeley) pp 251– (1965)
[14] The proceedings of the Bertrand Russell memorial conference (Uldum, 1971) pp 221– (1973)
[15] Theoria 35 pp 1– (1969)
[16] DOI: 10.2307/2266967 · Zbl 0039.00801 · doi:10.2307/2266967
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